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arXiv · 2503.02381

Secondary terms in the counting functions of quartic fields

Arul Shankar·Jacob Tsimerman
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Abstract

We prove that the smoothed counting function of the set of quartic fields, satisfying any finite set of local conditions, can be written as a linear combination of $X,X^{5/6}\log X,X^{5/6}$, upto an error term of $O(X^{13/16+o(1)})$. For certain sets of local conditions, namely, those cutting out ``$S_4$-families'' of quartic fields, we explicitly determine the leading constants of the secondary terms. We moreover express these constants in terms of secondary mass formulas associated to families of quartic fields

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Arul Shankar, Jacob Tsimerman. 2025-03-04. Secondary terms in the counting functions of quartic fields. https://arxiv.org/abs/2503.02381

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